Can they? definitely! Technically you don’t even need to be employed by them or be in the US for an employment based green card application [the application is based on a future bona fide job offer that you intend to accept when the green card is issued. What you do in the meantime is irrelevant as far as the application is concerned.] Do they? Depends on the company. Usually they come up with some reason like travel restrictions (which is not too much of an issue, since you get an Advance Parole soon after your application). Note that they can do both (H1 and green card) at the same time in parallel.
They applied for my GC when I was on my F1. My I-140 was also approved while I was on my F1
Yes
Yes. 100%
Absolutely they can and I know that LinkedIn does it. Good for you.
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You can apply for green card without h1, only downside is that it comes with travel restrictions. You would be advised to not travel outside United States till you get your h1. You should ask the lawyer for proper legal advice.